Question: Find the projection of $\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$ onto the plane $3x - y + 4z = 0.$
Let $P$ be the plane $3x - y + 4z = 0.$  We can take $\mathbf{n} = \begin{pmatrix} 3 \\ -1 \\ 4 \end{pmatrix}$ as the normal vector of plane $P.$

Let $\mathbf{v} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix},$ and let $\mathbf{p}$ be its projection onto plane $P.$  Note that $\mathbf{v} - \mathbf{p}$ is parallel to $\mathbf{n}.$

[asy]
import three;

size(160);
currentprojection = perspective(6,3,2);

triple I = (1,0,0), J = (0,1,0), K = (0,0,1);
triple O = (0,-0.5,0), V = (0,1.5,1), P = (0,1.5,0);

draw(surface((2*I + 2*J)--(2*I - 2*J)--(-2*I - 2*J)--(-2*I + 2*J)--cycle),paleyellow,nolight);
draw((2*I + 2*J)--(2*I - 2*J)--(-2*I - 2*J)--(-2*I + 2*J)--cycle);
draw((P + 0.1*(O - P))--(P + 0.1*(O - P) + 0.2*(V - P))--(P + 0.2*(V - P)));
draw(O--P,green,Arrow3(6));
draw(O--V,red,Arrow3(6));
draw(P--V,blue,Arrow3(6));
draw((1,-0.8,0)--(1,-0.8,0.2)--(1,-1,0.2));
draw((1,-1,0)--(1,-1,2),magenta,Arrow3(6));

label("$\mathbf{v}$", V, N, fontsize(10));
label("$\mathbf{p}$", P, S, fontsize(10));
label("$\mathbf{n}$", (1,-1,1), dir(180), fontsize(10));
label("$\mathbf{v} - \mathbf{p}$", (V + P)/2, E, fontsize(10));
[/asy]

Thus, $\mathbf{v} - \mathbf{p}$ is the projection of $\mathbf{v}$ onto $\mathbf{n}.$  Hence,
\[\mathbf{v} - \mathbf{p} = \frac{\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \cdot \begin{pmatrix} 3 \\ -1 \\ 4 \end{pmatrix}}{\begin{pmatrix} 3 \\ -1 \\ 4 \end{pmatrix} \cdot \begin{pmatrix} 3 \\ -1 \\ 4 \end{pmatrix}} \begin{pmatrix} 3 \\ -1 \\ 4 \end{pmatrix} = \frac{13}{26} \begin{pmatrix} 3 \\ -1 \\ 4 \end{pmatrix} = \begin{pmatrix} 3/2 \\ -1/2 \\ 2 \end{pmatrix}.\]Then
\[\mathbf{p} = \mathbf{v} - \begin{pmatrix} 3/2 \\ -1/2 \\ 2 \end{pmatrix} = \boxed{\begin{pmatrix} -1/2 \\ 5/2 \\ 1 \end{pmatrix}}.\]